Testing the limits of quantum mechanical superpositions

Abstract

Quantum physics has intrigued scientists and philosophers alike, because it challenges our notions of reality and locality — concepts that we have grown to rely on in our macroscopic world. It is an intriguing open question whether the linearity of quantum mechanics extends into the macroscopic domain. Scientific progress over the past decades inspires hope that this debate may be settled by table-top experiments.

Main

The past three decades have witnessed what has been termed1 the second quantum revolution: a renaissance of research on the quantum foundations, hand in hand with growing experimental capabilities2, revived the idea of exploiting quantum superpositions for technological applications, from information science3,4,5 to precision metrology6,7,8. Quantum mechanics has passed all precision tests with flying colours, but it still seems to be in conflict with our common sense. As quantum theory knows no boundaries, everything should fall under the sway of the superposition principle, including macroscopic objects. This is at the bottom of Schrödinger’s thought experiment of transforming a cat into a state that strikes us as classically impossible. And yet, ‘Schrödinger kittens’ of entangled photons9 and ions10 have been realized in the lab.

So why are the objects around us never found in superpositions of states that would be impossible in a classical description? One may emphasize the smallness of Planck’s constant, or point to decoherence theory, which describes how a system will effectively lose its quantum features when coupled to a quantum environment of sufficient size11,12. The formalism of decoherence, however, is based on the framework of unitary quantum mechanics, implying that some interpretational exercise is required not to become entangled in a multitude of parallel worlds13. More radically, one may ask whether quantum mechanics breaks down beyond a certain mass or complexity scale. As will be discussed below, such ideas can be motivated by the apparent incompatibility of quantum theory and general relativity. It is safe to state, in any case, that quantum superpositions of truly massive, complex objects are terra incognita. This makes them an attractive challenge for a growing number of sophisticated experiments.

We start by reviewing several prototypical tests of the superposition principle, focusing on the quantum states of motion exhibited by material objects. Particle position and momentum variables have a well-defined classical analogue, and they are therefore particularly suited to probe the macroscopic domain. We note that aspects of macroscopicity can also be addressed in experiments with photons14,15,16, with the phonons of ion chains17, and by squeezing pseudospins 8,18.

State of the art

Superconducting quantum interference devices (SQUIDs) have recently attracted a lot of interest, because they are promising elements of quantum information processing19. A SQUID is a superconducting loop segmented by Josephson junctions. Its electronic and transport properties are determined by a macroscopic wavefunction ordering the Cooper pairs. To exploit this macroscopicity it is appealing to consider a flux qubit20 (Fig. 1a): the single-valuedness of the wavefunction means that the magnetic flux encircled by a closed-loop supercurrent must be quantized. In particular, one can define a symmetric and an antisymmetric linear combination of two supercurrents, which circulate simultaneously in opposing directions. Billions of electrons may contribute coherently to the wavefunction over mesoscopic dimensions. The difference between the clockwise and anti-clockwise currents21 can reach about 2 μA, amounting to a local magnetic moment of about 1010 Bohr magnetons. This is an impressive number, which has led to the suggestion that SQUIDs may exhibit the most macroscopic quantum superposition to date. However, ‘only’ a few thousand of the Cooper pairs carrying the different currents are distinguishable22, which points to the need for an objective measure of macroscopicity (Box 1).

Figure 1: Superposition experiments.
figure1

a, A flux qubit realizes a quantum superposition of left- and right-circulating supercurrents21 with billions of electrons contributing to the quantum state. b, Neutron interferometry with perfect crystal beam-splitters holds the current record in matter-wave delocalization24, separating the quantum wave packet by up to 7 cm. c, Modern atom interferometry achieves coherence times beyond two seconds with wave-packet separations up to 1.5 cm (refs 36, 37, 38). d, Interference of two clouds of Bose–Einstein condensed diatomic lithium molecules101. e, Kapitza–Dirac–Talbot–Lau interferometer for macromolecules44,54,57. Figures reproduced with permission from: a, ref. 20, 2008 NPG; b, ref. 24, © 2002 Elsevier; d, ref. 101, © C. Kohstall and R. Grimm, University of Innsbruck, Austria; e, ref. 57, © 2010 RSC.

Historically, perfect-crystal neutron quantum optics23 made many interference experiments with atoms and photons possible. As the de Broglie wavelength of thermal neutrons is comparable to the lattice constant of silicon, quantum diffraction off the nuclei may split the neutron wavefunction at large angles. As of today, neutron interferometry still realizes the widest delocalization of any massive object24. With an arm separation up to 7 cm, enclosing an area of 80 cm2, it allows one to stick a hand between the two branches of a quantum state that describes a single microscopic particle (Fig. 1b). Even though neutrons are very light neutral particles, they are prime candidates for emergent tests of post-Newtonian gravity at short distances25,26. With an electrical polarizability twenty orders of magnitude smaller than for atoms, neutrons are much less sensitive to electrostatic perturbations, such as charges, patch effects or van der Waals forces.

Much better control and signal to noise can be achieved by using atoms. Atom interferometry (Fig. 1c) started about 30 years ago27,28,29. The development of Raman30 beam-splitters then transformed the tools of basic science into high-precision quantum sensors that split, invert and recombine the atomic wavefunction in three short laser pulses (Fig. 1c). In particular, inertial forces such as gravity and Coriolis forces31,32 have been measured with stunning precision in experiments that also promise new tests of general relativity33.

The mass in these experiments is always limited to that of a single atom, in practice to the caesium mass of 133 amu. A degree of macroscopicity can still be reached in the spatial extension of the wavefunction and in coherence time. The achievable delocalization depends on the momentum transfer in the beam-splitting element, whereas the coherence time is essentially determined by the duration of free fall in the apparatus. Both impressively wide-angle beam splitters34,35 and very long coherence times36 have been demonstrated separately, and been recently combined in an experiment with rubidium atoms, whose wave packets get separated for 2.3 s with a maximal distance of 1.4 cm (ref. 37). Future quantum sensors are expected to increase the sensitivity of quantum metrology by several orders of magnitude. The coherence time grows only with the square root of the device length, so that it will be practically limited to several seconds in Earth-bound devices, even in high-drop towers. Progress in matter-wave beam splitting will depend on improved wavefront control of the beam splitting lasers and other technological breakthroughs. If it were possible to build interferometers of 100 m length with beam-splitters capable of transferring a hundred grating momenta38, atomic matter would be delocalized over distances of metres. Even though designed for testing the effects of general relativity33,39, such experiments would also test the linearity of quantum mechanics40 as well as the homogeneity of spacetime41.

It is frequently suggested that ultra-cold atomic ensembles may serve to test the linearity of quantum physics even better, as all atoms can be described by a joint many-body wavefunction once they are cooled below the phase transition to Bose–Einstein condensation (Fig. 1d). Billions of non-interacting atoms may be united in a quantum degenerate state, which is, however, a product of single-particle states , so that interference of Bose-condensed atoms depends only on the de Broglie wavelength of single atoms. A genuinely entangled many-particle state akin to a Schrödinger cat state would be required to reduce the fringe spacing. Such macroscopic cat states with regard to the particle motion have remained an open challenge, even though entanglement in other degrees of freedom has been demonstrated between dozens of atoms7,8,42. In contrast to that, macromolecules and clusters open a new field involving strongly bound particles with internal temperatures up to 1,000 K. When N atoms are covalently linked into a single molecule they act as a single object in quantum interference experiments. The entire N-atom system is then delocalized over two or more interferometer arms.

Macromolecule interferometry started originally with the far-field diffraction of fullerenes43 and works with high-mass objects in currently two different settings: the Kapitza–Dirac–Tabot–Lau interferometer (KDTLI) and an all-optical interferometer in the time domain with pulsed ionization gratings (OTIMA). Both concepts were developed and implemented at the University of Vienna44,45 and are based on similar ideas. In high-mass matter-wave interference we face de Broglie wavelengths between 10 fm and 10 pm for objects between 1010 and 103amu. This is more than six orders of magnitude smaller than in all experiments with ultra-cold atoms. Macromolecules are not susceptible to established laser cooling techniques, although first steps towards the cavity cooling of 1010amu objects have been taken46,47. The particles therefore start out in rather mixed states, requiring near-field interference schemes48.

The KDTLI interferometer is sketched in Fig. 1e. It accepts a large variety of nanoparticles, because it uses only non-resonant gratings to split (G1), diffract (G2) and probe (G3) matter-waves. The first grating (G1) implements a spatially periodic transmission function. The size of the slits and the separation between G1 and G2 are chosen such that the position–momentum uncertainty in each slit is sufficient to expand each particle’s wavefunction to cover more than two slits in G2 downstream. To achieve this, G1 must be an absorptive mask, here realized as a silicon nitride nanostructure. Grating G2, a non-resonant standing light wave, imprints a spatially periodic phase onto the matter-wave. A near-field resonance effect rephases the wavefunctions to a molecular density pattern at the position of G3. Although one might capture the emerging quantum fringe pattern on a substrate for subsequent high-resolution microscopy49,50, it is often convenient to scan the absorptive mask G3 across the nanopattern: a plot of the number of transmitted particles as a function of the mask’s position reveals the molecular interferogram (Fig. 1e).

In contrast to the KDTLI, an OTIMA interferometer relies on three pulsed gratings that ionize and thus remove the molecules at the anti-nodes of an ultraviolet standing-wave laser beam51. Such all-optical gratings can handle of highly polarizable or polar particles, and their pulsed nature allows us to profit from working in the time domain. All particles exposed to the spatially extended nanosecond laser pulses then see the same grating for the same time, regardless of their velocity. This eliminates numerous dispersive dephasing phenomena, which is particularly beneficial for quantum tests at high masses52,53. KDTLI and OTIMA are ‘universal’ in the sense that they can accept a wide class of different objects and both avoid the detrimental effect of van der Waals forces in G2 by using non-resonant optical beam-splitters.

Experiments in the KDTLI currently hold the mass record in matter-wave interference, with a functionalized tetraphenylporphyrin molecule that combines 810 atoms into one particle with a molecular weight exceeding 10,000 amu (ref. 54). Even at an internal temperature of 500 K this object can be delocalized over a hundred times its own diameter and for more than 1 ms. Very recently, the OTIMA concept has been demonstrated45 with clusters of molecules. It will soon be used to explore quantum coherence at unprecedented masses52. Both interferometers also share a high potential for quantum-assisted metrology targeting internal properties, which reveal themselves in de Broglie experiments owing to the phase shift induced by external fields55,56,57.

Physics beyond the Schrödinger equation?

The experimental tests discussed so far confirm quantum mechanics impressively, as do high-precision spectroscopic measurements58,59 and tests of nonlocality60,61,62. Many physicists take for granted that quantum theory is valid on macroscopic scales, the more so because environmental decoherence explains why macroscopic objects seem to assume the classically distinguished states we observe in our everyday life11,12 (Fig. 2).

Figure 2: Accounting for environmental decoherence.
figure2

The theory of decoherence accounts for the impact of a quantum system on practically unobservable environmental degrees of freedom11,12. It can thus explain the effective super-selection of distinguished system states and the emergence of classical dynamics. From a practical point of view, decoherence theory tells us how strongly a quantum system must be isolated from its surroundings to be still expected to show quantum interference. The figure gives the ambient temperature and pressure requirements for observing OTIMA interference with gold clusters of 106,107 and108amu. Similarly demanding conditions for shielding environmental decoherence apply to the other described superposition tests. Figure adapted with permission from ref. 52, © 2011 APS.

Yet, there are good reasons to take seriously the possibility that quantum theory may fail beyond some scale. A compelling one is the difficulty of reconciling quantum theory with the nonlinear laws of general relativity, which treats spacetime as a dynamical entity. Most theories of quantum gravity suggest that there is a minimal observable length scale, often associated with the Planck length. One way to account for this phenomenologically is to postulate modified commutator relations for the canonical observables, which might be testable by monitoring the motion of massive pendulums at the quantum level63,64,65,66,67. The granularity of spacetime might manifest itself also in a fundamentally non-unitary time evolution of the quantum system, which would be observable as an intrinsic decoherence process41,68,69,70.

The alternative that gravity is not to be quantized, but fundamentally described by a classical field, suggests one should extend the Schrödinger equation nonlinearly to account for the gravitational self-interaction71,72. This idea is formalized in the Schrödinger–Newton equation, which can be obtained as the non-relativistic limit of self-gravitating Klein–Gordon fields73. It has been hypothesized that this equation defines the timescale and the basis states of a fundamental collapse mechanism. Indeed, an additional collapse-like stochastic process is required for any such nonlinear extension of the Schrödinger equation to ensure that the time evolution maps any initial state linearly to an ensemble described by a proper density operator. Otherwise an entangled particle pair would admit superluminal signalling — that is, violate causality — because the nonlinearity would imprint the basis of a distant measurement onto the reduced local state74. A gravitationally-inspired nonlinear modification of quantum mechanics75 can be made consistent with causality and observations at the price of a fictitiously large blurring of the involved mass density71.

The best studied nonlinear modification of quantum mechanics is the continuous spontaneous localization (CSL) model76,77. It augments the Schrödinger equation for elementary particles with a Gaussian noise term that gives rise to a continuous stochastic collapse of wavefunctions delocalized beyond about 100 nm. The origin of the stochastic process remains unspecified; one may view it either as a fundamental trait of nature, or as the repercussion of an inaccessible underlying dynamics 78. The CSL effect would be very weak and practically unobservable on the atomic level, but it would get strongly amplified for bound atoms forming a solid, such as the pointer of a measurement device. Any superposition of macroscopically distinct positions would rapidly collapse, in agreement with Born’s rule, to a ‘classical’ state characterized by a localized, objective wavefunction. This way the model serves its purpose of restoring objective classical reality on the scale of everyday objects, allowing one to dispense with the measurement postulate.

It is a contentious issue whether such macrorealism79 is required in a plausible description of physical reality. Independent of that, the CSL model serves as a cautionary tale. It proves that there are competing descriptions of nature, which predict strongly different effects at macroscopic scales, even though they are compatible with all experiments and cosmological observations carried out so far71,80. One may invoke metaphysical arguments in favour of one or another theory, but empirically their status is equal, and only future experiments will be able to tell them apart.

Venturing towards macroscopic quantum superpositions

Various different systems have been suggested for probing the quantum superposition principle at mesoscopic or even macroscopic scales. This raises the question how to objectively assess the degree of macroscopicity reached in different experiments40 (Box 1).

The gravitational collapse hypothesis81 inspired a proposal to create a quantum superposition in the centre-of-mass motion of a micromirror82 (Fig. 3a). A lightweight (picogram) mirror suspended from a cantilever can close a cavity acting as one arm of a Michelson interferometer. A single photon entering the interferometer excites a superposition of the two cavity modes. The radiation pressure of the single photon induces a deflective oscillation of the small mirror by approximately the width of the zero-point motion. Which-path information is thus left behind once the photon escapes from the cavities, unless this occurs at a multiple of the cantilever oscillation period, when the original state of the mirror reappears. Observing the recurrence of optical interference after one such oscillation period would therefore prove that the mirror was in a superposition state82,83.

Figure 3: Interference schemes for large masses.
figure3

a, The superposition of a micromechanical oscillator can be triggered by scattering a single photon in a Michelson interferometer. b, Time-domain matter-wave interferometry of nanoparticles with pulsed laser gratings is expected to be scalable to high masses. c, Far-field interference of nanospheres at a measurement-induced double slit may be observed by correlating the detected positions with a phase measurement.

This is a difficult experiment because a relatively massive oscillator with an eigenfrequency in the low kilohertz regime is required for probing gravitational collapse. This implies that the oscillator ground state is reached only at microkelvin temperatures. Ground-state cooling is easier with lighter and more rigid megahertz or gigahertz oscillators, and by addressing normal modes with stronger opto-mechanical coupling. This feat has been achieved recently with the flexural mode of a circular aluminium micro-membrane using optical side-band cooling84,85. Many groups worldwide have embarked on studying such nanomechanical oscillators86, which can serve as an interface between quantum systems. However, it has been difficult to observe genuine quantum effects in optomechanical systems because they still lack the strong nonlinear coupling required to generate quantum states of motion that differ qualitatively from classical ones. As a first step in this direction a piezoelectric resonator was coupled coherently to a superconducting loop87.

The distinctive feature of micromechanical devices compared with other quantum systems is their very high mass. However, the quantum delocalization of the oscillatory ground state, which is a collective degree of freedom involving all the atoms, will reach at most about one picometre in conceivable set-ups—a tiny fraction of the size of an atom. This indicates why some matter-wave experiments will reach beyond the macroscopicity of a possible superposition of the micro-membrane (Box 1).

As any clamped nanostructure will be prone to damping, recent proposals88,89,90 consider levitating dielectric nanoparticles in the focus of an intense laser beam. Cooling the centre-of-mass motion to the ground state should be feasible, owing to their lower mass and the high trap frequencies. Moreover, the nanosphere position can be coupled nonlinearly to a resonator light field by placing the optical trap at the node of a Fabry–Pérot cavity. This opens the possibility to create distinctively non-classical states, and to probe the wave nature of the nano-spheres, for example, by implementing an effective double-slit91. In this scheme one would drop the nanosphere once it has been cooled to the ground state of a dipole trap. After the wave packet is sufficiently dispersed, a laser pulse passing through a Fabry–Pérot cavity reveals the square of the position by a homodyne measurement of the cavity light field. One thus learns the distance of the sphere from the cavity centre, but not whether it is on the left or right, thus effectively projecting its wavefunction to a spatial superposition state. An interference pattern should then be observable after a further free evolution of the sphere, and after many repetitions, if one correlates the detected positions with the results of the homodyne measurements (Fig. 3b). The nanosphere position would be delocalized by approximately the diameter of the sphere, which should be sufficiently large to test the effects of the CSL collapse model.

A straightforward strategy for probing the wave nature of nanometre-sized objects is to push established matter-wave interference schemes to the limits of large masses. The OTIMA interferometer (Fig. 3c) should allow us to probe the quantum nature of 105amu particles if the source ejects them with a velocity of about 10 m s−1 (ref. 53). Objects with a diameter up to 10 nm would get delocalized over 80 nm. In the future, even nanoparticles in the mass range of 108amu might be diffracted with an OTIMA scheme, for example gold clusters with a diameter of 22 nm. Successful interference at these masses would falsify all current CSL predictions52. However, it would require us to counteract the gravitational acceleration, by noise-free levitation techniques or by going to a microgravity environment, to allow the wavefunction to expand over a coherence time of many seconds. Moreover, environmental decoherence would need to be suppressed by setting the ambient pressure to below 10−11 mbar and by cooling the apparatus to cryogenic temperatures92; (Fig. 2). The biggest challenge, both for OTIMA interferometry and the realization of a projective double slit, is the preparation of size-selected neutral particles in ultra-high vacuum at low internal and motional temperatures. Some promising first steps have been achieved by recent demonstrations of optical feedback cooling93,94 and cavity cooling46,47.

Perspectives

Will the quantum superposition principle stand the test of time? We have emphasized that this question is neither crazy nor heretical. Objective modifications of quantum mechanics can be set up that agree with all observations and experiments so far, while describing a tangible breakdown of quantum theory at the macroscale. Whether quantum mechanics is universally valid is thus not an issue of conviction or metaphysical reasoning, but an empirical question, to be answered only by future experiments.

A great variety of quantum systems may be used to demonstrate mechanical superposition states, whose mass, geometric size and delocalization scales may vary by orders of magnitude. Any such quantum test, if carried out successfully, will rule out a generic class of objective modifications of quantum mechanics. Using the scope of this falsified class as a yardstick, it is remarkable that totally different experimental approaches lead to comparable degrees of macroscopicity This suggests that there is no single golden strategy to be pursued, and much will depend on experimental advances and ideas. It is thus a long and exciting journey into the realm of large quantum superpositions, and one worth taking.

References

  1. 1

    Dowling, J. P. & Milburn, G. J. Quantum technology: The second quantum revolution. Phil. Trans. A 361, 1655–1674 (2003).

    ADS  MathSciNet  Google Scholar 

  2. 2

    Zeilinger, A. Experiment and the foundations of quantum physics. Rev. Mod. Phys. 71, S288–S297 (1999).

    Google Scholar 

  3. 3

    Trabesinger, A. Quantum simulation. Nature Phys. 8, 263–263 (2012).

    ADS  Google Scholar 

  4. 4

    Bennett, C. H. & DiVincenzo, D. P. Quantum information and computation. Nature 404, 247–255 (2000).

    ADS  MATH  Google Scholar 

  5. 5

    Southwell, K. Quantum coherence. Nature 453, 1003–1003 (2008).

    ADS  Google Scholar 

  6. 6

    Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nature Phys. 5, 222–229 (2011).

    ADS  Google Scholar 

  7. 7

    Riedel, M. F. et al. Atom-chip-based generation of entanglement for quantum metrology. Nature 464, 1170–1173 (2010).

    ADS  Google Scholar 

  8. 8

    Gross, C., Zibold, T., Nicklas, E., Estève, J. & Oberthaler, M. K. Nonlinear atom interferometer surpasses classical precision limit. Nature 464, 1165–1169 (2010).

    ADS  Google Scholar 

  9. 9

    Haroche, S. Nobel Lecture: Controlling photons in a box and exploring the quantum to classical boundary. Rev. Mod. Phys. 85, 1083–1102 (2013).

    ADS  Google Scholar 

  10. 10

    Wineland, D. J. Nobel Lecture: Superposition, entanglement, and raising Schrödinger’s cat. Rev. Mod. Phys. 85, 1103–1114 (2013).

    ADS  Google Scholar 

  11. 11

    Joos, E. et al. Decoherence and the Appearance of a Classical World in Quantum Theory 2nd edition (Springer, 2003).

    Google Scholar 

  12. 12

    Zurek, W. H. Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715–775 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  13. 13

    Laloë, F. Do We Really Understand Quantum Mechanics? (Cambridge Univ. Press, 2012).

    Google Scholar 

  14. 14

    Fickler, R. et al. Quantum entanglement of high angular momenta. Science 338, 640–643 (2012).

    ADS  Google Scholar 

  15. 15

    Ma, X. S. et al. Quantum teleportation over 143 kilometres using active feed-forward. Nature 489, 269–273 (2012).

    ADS  Google Scholar 

  16. 16

    Kirchmair, G. et al. Observation of quantum state collapse and revival due to the single-photon Kerr effect. Nature 495, 205–209 (2013).

    ADS  Google Scholar 

  17. 17

    Monz, T. et al. 14-qubit entanglement: Creation. Phys. Rev. Lett. 106, 130506 (2011).

    ADS  Google Scholar 

  18. 18

    Julsgaard, B., Kozhekin, A. & Polzik, E. S. Experimental long-lived entanglement of two macroscopic objects. Nature 413, 400–403 (2001).

    ADS  Google Scholar 

  19. 19

    Devoret, M. H. & Schoelkopf, R. J. Superconducting circuits for quantum information: An outlook. Science 339, 1169–1174 (2013).

    ADS  Google Scholar 

  20. 20

    Clarke, J. & Wilhelm, F. K. Superconducting quantum bits. Nature 453, 1031–1042 (2008).

    ADS  Google Scholar 

  21. 21

    Friedman, J., Patel, V., Chen, W., Tolpygo, S. & Lukens, J. Quantum superposition of distinct macroscopic states. Nature 406, 43–46 (2000).

    ADS  Google Scholar 

  22. 22

    Korsbakken, J., Wilhelm, F. & Whaley, K. The size of macroscopic superposition states in flux qubits. Europhys. Lett. 89, 30003 (2010).

    ADS  Google Scholar 

  23. 23

    Rauch, H., Treimer, W. & Bonse, U. Test of a single crystal neutron interferometer. Phys. Rev. A 47, 369–371 (1974).

    Google Scholar 

  24. 24

    Zawisky, M., Baron, M., Loidl, R. & Rauch, H. Testing the world’s largest monolithic perfect crystal neutron interferometer. Nucl. Instrum. Methods Phys. Res. A 481, 406–413 (2002).

    ADS  Google Scholar 

  25. 25

    Nesvizhevsky, V. V. et al. Quantum states of neutrons in the earth’s gravitational field. Nature 415, 298–300 (2002).

    ADS  Google Scholar 

  26. 26

    Jenke, T., Geltenbort, P., Lemmel, H. & Abele, H. Realization of a gravity–resonance–spectroscopy technique. Nature Phys. 7, 468–472 (2011).

    ADS  Google Scholar 

  27. 27

    Gould, P. L., Ruff, G. A. & Pritchard, D. E. Diffraction of atoms by light: The near-resonant Kapitza–Dirac effect. Phys. Rev. Lett. 56, 827–830 (1986).

    ADS  Google Scholar 

  28. 28

    Keith, D. W., Schattenburg, M. L., Smith, H. I. & Pritchard, D. E. Diffraction of atoms by a transmission grating. Phys. Rev. Lett. 61, 1580–1583 (1988).

    ADS  Google Scholar 

  29. 29

    Bordé, C. Atomic interferometry with internal state labelling. Phys. Lett. A 140, 10–12 (1989).

    ADS  Google Scholar 

  30. 30

    Kasevich, M. & Chu, S. Atomic interferometry using stimulated Raman transitions. Phys. Rev. Lett. 67, 181–184 (1991).

    ADS  Google Scholar 

  31. 31

    Peters, A., Yeow-Chung, K. & Chu, S. Measurement of gravitational acceleration by dropping atoms. Nature 400, 849–852 (1999).

    ADS  Google Scholar 

  32. 32

    Stockton, J. K., Takase, K. & Kasevich, M. A. Absolute geodetic rotation measurement using atom interferometry. Phys. Rev. Lett. 107, 133001 (2011).

    ADS  Google Scholar 

  33. 33

    Hohensee, M., Chu, S., Peters, A. & Müller, H. Equivalence principle and gravitational redshift. Phys. Rev. Lett. 106, 151102 (2011).

    ADS  Google Scholar 

  34. 34

    Müller, H., Chiow, S-w., Long, Q., Herrmann, S. & Chu, S. Atom interferometry with up to 24-photon-momentum-transfer beam splitters. Phys. Rev. Lett. 100, 180405 (2008).

    ADS  Google Scholar 

  35. 35

    Chiow, S., Kovachy, T., Chien, H. & Kasevich, M. 102k large area atom interferometers. Phys. Rev. Lett. 107, 130403 (2011).

    ADS  Google Scholar 

  36. 36

    Müntinga, H. et al. Interferometry with Bose–Einstein condensates in microgravity. Phys. Rev. Lett. 110, 093602 (2013).

    ADS  Google Scholar 

  37. 37

    Dickerson, S. M., Hogan, J. M., Sugarbaker, A., Johnson, D. M. S. & Kasevich, M. A. Multiaxis inertial sensing with long-time point source atom interferometry. Phys. Rev. Lett. 111, 083001 (2013).

    ADS  Google Scholar 

  38. 38

    Dimopoulos, S., Graham, P., Hogan, J. & Kasevich, M. Testing general relativity with atom interferometry. Phys. Rev. Lett. 98, 1–4 (2007).

    Google Scholar 

  39. 39

    Bouyer, P. & Landragin, A. Interférométrie atomique et gravitation: du sol à l’espace. Journées de l’action spécifique GRAM (Gravitation, Références, Astronomie, Métrologie) (Nice, France, 2010).

  40. 40

    Nimmrichter, S. & Hornberger, K. Macroscopicity of mechanical quantum superposition states. Phys. Rev. Lett. 110, 160403 (2013).

    ADS  Google Scholar 

  41. 41

    Percival, I. C. & Strunz, W. T. Detection of spacetime fluctuation by a model interferometer. Proc. R. Soc. Lond. A 453, 431–446 (1997).

    ADS  Google Scholar 

  42. 42

    Sherson, J. et al. Quantum teleportation between light and matter. Nature 443, 557–560 (2006).

    ADS  Google Scholar 

  43. 43

    Arndt, M. et al. Wave-particle duality of C60 molecules. Nature 401, 680–682 (1999).

    ADS  Google Scholar 

  44. 44

    Gerlich, S. et al. A Kapitza–Dirac–Talbot–Lau interferometer for highly polarizable molecules. Nature Phys. 3, 711–715 (2007).

    ADS  Google Scholar 

  45. 45

    Haslinger, P. et al. A universal matter-wave interferometer with optical ionization gratings in the time domain. Nature Phys. 9, 144–148 (2013).

    ADS  Google Scholar 

  46. 46

    Kiesel, N. et al. Cavity cooling of an optically levitated nanoparticle. Proc. Natl Acad. Sci. USA 110, 14180–14185 (2013).

    ADS  Google Scholar 

  47. 47

    Asenbaum, P., Kuhn, S., Nimmrichter, S., Sezer, U. & Arndt, M. Cavity cooling of free silicon nanoparticles in high-vacuum. Nature Commun. 4, 2743 (2013).

    ADS  Google Scholar 

  48. 48

    Clauser, J. in Experimental Metaphysics (eds Cohen, R. S., Horne, M. & Stachel, J.) 1–11 (Kluwer Academic, 1997).

    Google Scholar 

  49. 49

    Juffmann, T. et al. Wave and particle in molecular interference lithography. Phys. Rev. Lett. 103, 263601 (2009).

    ADS  Google Scholar 

  50. 50

    Juffmann, T. et al. Real-time single-molecule imaging of quantum interference. Nature Nanotech. 7, 297–300 (2012).

    ADS  Google Scholar 

  51. 51

    Reiger, E., Hackermüller, L., Berninger, M. & Arndt, M. Exploration of gold nanoparticle beams for matter wave interferometry. Opt. Commun. 264, 326–332 (2006).

    ADS  Google Scholar 

  52. 52

    Nimmrichter, S., Hornberger, K., Haslinger, P. & Arndt, M. Testing spontaneous localization theories with matter-wave interferometry. Phys. Rev. A 83, 043621 (2011).

    ADS  Google Scholar 

  53. 53

    Nimmrichter, S., Haslinger, P., Hornberger, K. & Arndt, M. Concept of an ionizing time-domain matter-wave interferometer. New J. Phys. 13, 075002 (2011).

    ADS  Google Scholar 

  54. 54

    Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M. & Tüxen, J. Matter-wave interference of particles selected from a molecular library with masses exceeding 10 000 amu. Phys. Chem. Chem. Phys. 15, 14696–14700 (2013).

    Google Scholar 

  55. 55

    Berninger, M., Stéfanov, A., Deachapunya, S. & Arndt, M. Polarizability measurements in a molecule near-field interferometer. Phys. Rev. A 76, 013607 (2007).

    ADS  Google Scholar 

  56. 56

    Gerlich, S. et al. Matter-wave metrology as a complementary tool for mass spectrometry. Angew. Chem-Int. Ed. 47, 6195–6198 (2008).

    Google Scholar 

  57. 57

    Tüxen, J., Gerlich, S., Eibenberger, S., Arndt, M. & Mayor, M. De Broglie interference distinguishes between constitutional isomers. Chem. Commun. 46, 4145–4147 (2010).

    Google Scholar 

  58. 58

    Niering, M. et al. Measurement of the hydrogen 1S- 2S transition frequency by phase coherent comparison with a microwave cesium fountain clock. Phys. Rev. Lett. 84, 5496–5499 (2000).

    ADS  Google Scholar 

  59. 59

    Odom, B., Hanneke, D., D’Urso, B. & Gabrielse, G. New measurement of the electron magnetic moment using a one-electron quantum cyclotron. Phys. Rev. Lett. 97, 030801 (2006).

    ADS  Google Scholar 

  60. 60

    Freedman, S. J. & Clauser, J. F. Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938–941 (1972).

    ADS  Google Scholar 

  61. 61

    Aspect, A., Dalibard, J. & Roger, G. Experimental test of Bell’s inequalities using time- varying analyzers. Phys. Rev. Lett. 49, 1804–1807 (1982).

    ADS  MathSciNet  Google Scholar 

  62. 62

    Giustina, M. et al. Bell violation with entangled photons, free of the fair-sampling assumption. Nature 497, 227–230 (2013).

    ADS  Google Scholar 

  63. 63

    Abbott, B. et al. Observation of a kilogram-scale oscillator near its quantum ground state. New J. Phys. 11, 073032 (2009).

    ADS  Google Scholar 

  64. 64

    Das, S. & Vagenas, E. C. Universality of quantum gravity corrections. Phys. Rev. Lett. 101, 221301 (2008).

    ADS  Google Scholar 

  65. 65

    Bojowald, M. & Kempf, A. Generalized uncertainty principles and localization of a particle in discrete space. Phys. Rev. D 86, 085017 (2012).

    ADS  Google Scholar 

  66. 66

    Pikovski, I., Vanner, M. R., Aspelmeyer, M., Kim, M. & Brukner, Č. Probing Planck-scale physics with quantum optics. Nature Phys. 8, 393–397 (2012).

    ADS  Google Scholar 

  67. 67

    Marin, F. et al. Gravitational bar detectors set limits to Planck-scale physics on macroscopic variables. Nature Phys. 9, 71–73 (2012).

    ADS  Google Scholar 

  68. 68

    Gambini, R., Porto, R. A. & Pullin, J. Realistic clocks, universal decoherence, and the black hole information paradox. Phys. Rev. Lett. 93, 240401 (2004).

    ADS  MathSciNet  Google Scholar 

  69. 69

    Milburn, G. J. Lorentz invariant intrinsic decoherence. New J. Phys. 8, 96 (2006).

    ADS  Google Scholar 

  70. 70

    Wang, C. H-T., Bingham, R. & Mendonça, J. T. Quantum gravitational decoherence of matter waves. Class. Quantum Gravity 23, L59–L65 (2006).

    MathSciNet  MATH  Google Scholar 

  71. 71

    Bassi, A., Lochan, K., Satin, S., Singh, T. P. & Ulbricht, H. Models of wave-function collapse, underlying theories, and experimental tests. Rev. Mod. Phys. 85, 471–527 (2013).

    ADS  Google Scholar 

  72. 72

    Yang, H., Miao, H., Lee, D-S., Helou, B. & Chen, Y. Macroscopic quantum mechanics in a classical spacetime. Phys. Rev. Lett. 110, 170401 (2013).

    ADS  Google Scholar 

  73. 73

    Giulini, D. & Großardt, A. The Schrödinger-Newton equation as a non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields. Class. Quantum Gravity 29, 215010 (2012).

    ADS  MATH  Google Scholar 

  74. 74

    Gisin, N. Stochastic quantum dynamics and relativity. Helv. Phys. Acta 62, 363–371 (1989).

    MathSciNet  Google Scholar 

  75. 75

    Diósi, L. A universal master equation for the gravitational violation of quantum mechanics. Phys. Lett. A 120, 377–381 (1987).

    ADS  MathSciNet  Google Scholar 

  76. 76

    Ghirardi, G. C., Pearle, P. & Rimini, A. Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles. Phys. Rev. A 42, 78–89 (1990).

    ADS  MathSciNet  Google Scholar 

  77. 77

    Bassi, A. & Ghirardi, G. Dynamical reduction models. Phys. Rep. 379, 257–426 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  78. 78

    Adler, S. L. Quantum Theory as an Emergent Phenomenon (Cambridge Univ. Press, 2004).

    Google Scholar 

  79. 79

    Leggett, A. J. Testing the limits of quantum mechanics: Motivation, state of play, prospects. J. Phys. Condens. Mater. 14, R415–R451 (2002).

    ADS  Google Scholar 

  80. 80

    Feldmann, W. & Tumulka, R. Parameter diagrams of the GRW and CSL theories of wavefunction collapse. J. Phys. A 45, 065304 (2012).

    ADS  MathSciNet  MATH  Google Scholar 

  81. 81

    Penrose, R. On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28, 581–600 (1996).

    ADS  MathSciNet  MATH  Google Scholar 

  82. 82

    Marshall, W., Simon, C., Penrose, R. & Bouwmeester, D. Towards quantum superpositions of a mirror. Phys. Rev. Lett. 91, 130401 (2003).

    ADS  MathSciNet  Google Scholar 

  83. 83

    Bose, S., Jacobs, K. & Knight, P. Scheme to probe the decoherence of a macroscopic object. Phys. Rev. A 59, 3204–3210 (1999).

    ADS  Google Scholar 

  84. 84

    Teufel, J. D. et al. Sideband cooling of micromechanical motion to the quantum ground state. Nature 475, 359–363 (2011).

    ADS  Google Scholar 

  85. 85

    Chan, J. et al. Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature 478, 89–92 (2011).

    ADS  Google Scholar 

  86. 86

    Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Preprint at http://arxiv.org/abs/1303.0733 (2013).

  87. 87

    O’Connell, A. D. et al. Quantum ground state and single-phonon control of a mechanical resonator. Nature 464, 697–703 (2010).

    ADS  Google Scholar 

  88. 88

    Chang, D. E. et al. Cavity opto-mechanics using an optically levitated nanosphere. Proc. Natl Acad. Sci. USA 107, 1005–1010 (2010).

    ADS  Google Scholar 

  89. 89

    Romero-Isart, O., Juan, M. L., Quidant, R. & Cirac, J. I. Toward quantum superposition of living organisms. New J. Phys. 12, 033015 (2010).

    ADS  Google Scholar 

  90. 90

    Barker, P. F. & Shneider, M. N. Cavity cooling of an optically trapped nanoparticle. Phys. Rev. A 81, 023826 (2010).

    ADS  Google Scholar 

  91. 91

    Romero-Isart, O. et al. Large quantum superpositions and interference of massive nanometer-sized objects. Phys. Rev. Lett. 107, 020405 (2011).

    ADS  Google Scholar 

  92. 92

    Hornberger, K., Gerlich, S., Haslinger, P., Nimmrichter, S. & Arndt, M. Colloquium: Quantum interference of clusters and molecules. Rev. Mod. Phys. 84, 157–173 (2012).

    ADS  Google Scholar 

  93. 93

    Li, T., Kheifets, S. & Raizen, M. G. Millikelvin cooling of an optically trapped microsphere in vacuum. Nature Phys. 7, 527–530 (2011).

    ADS  Google Scholar 

  94. 94

    Gieseler, J., Deutsch, B., Quidant, R. & Novotny, L. Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. Phys. Rev. Lett. 109, 103603 (2012).

    ADS  Google Scholar 

  95. 95

    Dür, W., Simon, C. & Cirac, J. I. Effective size of certain macroscopic quantum superpositions. Phys. Rev. Lett. 89, 210402 (2002).

    ADS  Google Scholar 

  96. 96

    Björk, G. & Mana, P. A size criterion for macroscopic superposition states. J. Opt. B 6, 429–436 (2004).

    ADS  Google Scholar 

  97. 97

    Korsbakken, J. I., Whaley, K. B., Dubois, J. & Cirac, J. I. Measurement-based measure of the size of macroscopic quantum superpositions. Phys. Rev. A 75, 042106 (2007).

    ADS  Google Scholar 

  98. 98

    Marquardt, F., Abel, B. & von Delft, J. Measuring the size of a quantum superposition of many-body states. Phys. Rev. A 78, 012109 (2008).

    ADS  Google Scholar 

  99. 99

    Lee, C-W. & Jeong, H. Quantification of macroscopic quantum superpositions within phase space. Phys. Rev. Lett. 106, 220401 (2011).

    ADS  Google Scholar 

  100. 100

    Fröwis, F. & Dür, W. Measures of macroscopicity for quantum spin systems. New J. Phys. 14, 093039 (2012).

    ADS  Google Scholar 

  101. 101

    Kohstall, C. et al. Observation of interference between two molecular Bose–Einstein condensates. New J. Phys. 13, 065027 (2011).

    ADS  Google Scholar 

Download references

Acknowledgements

We thank S. Nimmrichter for helpful discussions, and we acknowledge support by the European Commission within NANOQUESTFIT (No. 304886). M.A. is supported by the Austrian FWF (Wittgenstein Z149-N16) and by the ERC (AdvG 320694 Probiotiqus), K.H. by the DFG (HO 2318/4-1 and SFB/TR12). We thank the WE Heraeus Foundation for supporting the physics school ‘Exploring the Limits of the Quantum Superposition Principle’.

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Markus Arndt or Klaus Hornberger.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Arndt, M., Hornberger, K. Testing the limits of quantum mechanical superpositions. Nature Phys 10, 271–277 (2014). https://doi.org/10.1038/nphys2863

Download citation

Further reading

Search

Quick links

Sign up for the Nature Briefing newsletter for a daily update on COVID-19 science.
Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing